#2.1
Know and be able to use vocabulary and notation appropriate to the subject at this level.
Vocabulary includes constant, coefficient, expression, equation, function, identity, index, term, variable, unknown.
Notation: \(f(x)\)
#2.2
Be able to solve linear equations in one unknown.
Including those containing brackets, fractions and the unknown on both sides of the equation.
#2.3
Be able to change the subject of a formula.
Including cases where the new subject appears on both sides of the original formula, and cases involving squares, square roots and reciprocals.
#2.4
Be able to solve quadratic equations.
By factorising, completing the square, using the formula and graphically. Includes quadratic equations in a function of the unknown.
#2.5
Be able to find the discriminant of a quadratic function and understand its significance.
The condition for distinct real roots of \(ax^2 + bx + c = 0\) is: Discriminant \(> 0\).
The condition for repeated roots is: Discriminant \(= 0\).
The condition for no real roots is: Discriminant \(< 0\).
Notation: For \(ax^2 + bx + c = 0\) the discriminant is \(b^2 - 4ac\).
[Excludes: Complex roots.]
#2.6
Be able to solve linear simultaneous equations in two unknowns.
By elimination and by substitution.
#2.7
Be able to solve simultaneous equations in two unknowns with one equation linear and one quadratic.
By elimination and by substitution.
#2.8
Know the significance of points of intersection of two graphs with relation to the solution of equations.
Including simultaneous equations.
#2.9
Be able to solve linear inequalities in one variable.
Be able to represent and interpret linear inequalities graphically e.g. \(y > x + 1\).
Including those containing brackets and fractions.
#2.10
Be able to solve quadratic inequalities in one variable.
Be able to represent and interpret quadratic inequalities graphically e.g. \(y > ax^2 + bx + c\).
Algebraic and graphical treatment of solution of quadratic inequalities.
For regions defined by inequalities learners must state clearly which regions are included and whether the boundaries are included. No particular shading convention is expected.
#2.11
Be able to express solutions of inequalities through correct use of ‘and’ and ‘or’, or by using set notation.
Learners will be expected to express solutions to quadratic inequalities in an appropriate version of one of the following ways.
\(x ≤ 1\) or \(x ≥ 4\)
\(\{x:x ≤ 1\} ∪ \{x:ax ≥ 4\}\)
\(2 < x < 5\)
\(x < 5\) and \(x > 2\)
\(\{x:x < 5\} ∩ \{x:x > 2\}\)
Notation: \(\{x : x > 4\}\)
#2.13
Be able to rationalise the denominator of a surd.
e.g. \(\dfrac{1}{5+\sqrt{3}} = \dfrac{5-\sqrt{3}}{22}\)
#2.14
Understand and be able to use the laws of indices for all rational exponents.
\(x^a × x^b = x^{a+b}\), \(x^a ÷ x^b = x^{a−b}\), \((x^a)^n = x^{an}\)
#2.15
Understand and be able to use negative, fractional and zero indices.
\(x^{-a}=\dfrac{1}{x^a}\), \(x^0=1 (x \neq 0)\), \(x^{\frac{1}{a}} = \sqrt[a]{x}\)
#2.16
Understand and use proportional relationships and their graphs.
For one variable directly or inversely proportional to a power or root of another.
#2.17
Be able to express algebraic fractions as partial fractions.
Fractions with constant or linear numerators and denominators up to three linear terms. Includes squared linear terms in denominator.
[Excludes: Fractions with a quadratic or cubic which cannot be factorised in the denominator.]
#2.18
Be able to simplify rational expressions.
Including factorising, cancelling and simple algebraic division. Any correct method of algebraic division may be used.
[Excludes: Division by non-linear expressions.]